Pushout of spaces

Question

Suppose that we have a map between two pushout diagrams of topological spaces $$ [A_{1}\leftarrow A_{0}\rightarrow A_{2} ] \rightarrow [B_{1}\leftarrow B_{0}\rightarrow B_{2} ]$$ such that for any $i\in\{0,1,2\}$, $A_{i}\rightarrow B_{i}$ is a trivial cofibration of topological spaces. And suppose that all spaces are CW-complexes. Moreover all maps $A_{i}\rightarrow A_{j}$ and $B_{i}\rightarrow B_{j}$ are supposed to be surjective. What can we say about the induced map $$ colim [A_{1}\leftarrow A_{0}\rightarrow A_{2} ] \rightarrow colim[B_{1}\leftarrow B_{0}\rightarrow B_{2} ]$$ e.g. is it a weak equivalence ?

Answer 1

Take $B_0$ to be an annulus. Let $B_1$ be the space obtained by collapsing the left half of the inner circle to a point, and let $B_2$ be obtained by collapsing the right half of the inner circle to a point. Let $A_0$, $A_1$ and $A_2$ all be the outer circle of $B_0$. Then the $A$-pushout is a circle whereas the $B$-pushout is a disc.

You could ask more questions along these lines, and it might take some ingenuity to find counterexamples, but they will always be there. You will almost never have good homotopical control over pushouts unless at least one arm of the pushout is a cofibration, or you have very specific information about the spaces involved.

Answer 2

When I see questions like this, I always reach for Dwyer-Spalinski "Homotopy theories and model categories." I just saw another question like this a couple of days ago, and a counterexample was given to show the colimit is not the homotopy colimit. I'll give a general answer to help whoever is asking these questions.

Consider the projective model structure on diagrams $C^D$ where $C$ is a model category (for you, it's Top) and $D$ is the category $a\gets b \to c$. You have a morphism $A\to B$ in $C^D$, that looks plausibly like a trivial cofibration, and you want to know when the colimit is a weak equivalence.

Proposition 10.6 in Dwyer-Spalinski characterizes the cofibrations of the projective model structure. One way for you to proceed would be to prove your morphism $A\to B$ is a projective trivial cofibration. According to Dwyer-Spalinski, that means it's a weak equivalence objectwise, and the following three maps are cofibrations: (1) $A_0\to B_0$ (2) the map from the pushout of $B_0 \gets A_0 \to A_1$ to $B_1$ (3) the map from the pushout of $B_0 \gets A_0 \to A_2$ to $B_2$ Offhand, it's not obvious to me that these pushout maps are cofibrations of spaces.

Another way would be to use Ken Brown's lemma, since $A\to B$ is a weak equivalence. So, you would need to prove $A$ and $B$ are projectively cofibrant. It's not enough that the spaces $A_i$ and $B_i$ are CW complexes. You also need that the maps $A_0\to A_1$, etc, are cofibrations. In this case, the colimit is the homotopy colimit, so the induced map on colimits is a weak equivalence.

It's not clear to me how the surjections you have help. If they were injections that would be great. Without some cofibrancy condition on the maps $A_i\to A_j$ and $B_i\to B_j$, I think you won't get a positive answer.